Asymptotics of sharp constants of Markov-Bernstein inequalities in integral norm with Jacobi weight

Abstract:

The classical A. Markov inequality establishes a relation between the maximum modulus or the $L^{\infty }\left ([-1,1]\right )$ norm of a polynomial $Q_{n}$ and of its derivative: $\|Q’_{n}\|\leqslant M_{n} n^{2}\|Q_{n}\|$, where the constant $M_{n}=1$ is sharp. The limiting behavior of the sharp constants $M_{n}$ for this inequality, considered in the space $L^{2}\left ([-1,1], w^{(\alpha ,\beta )}\right )$ with respect to the classical Jacobi weight $w^{(\alpha ,\beta )}(x):=(1-x)^{\alpha }(x+1)^{\beta }$, is studied. We prove that, under the condition $|\alpha - \beta | < 4$, the limit is $\lim _{n \to \infty } M_{n} = 1/(2 j_{\nu })$ where $j_{\nu }$ is the smallest zero of the Bessel function $J_{\nu }(x)$ and $2 \nu =\mbox {min}(\alpha , \beta ) - 1$.